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Evolution strategies
Chapter 4
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES quick overview
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Developed: Germany in the 1970’s
Early names: I. Rechenberg, H.-P. Schwefel
Typically applied to:
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Attributed features:
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numerical optimisation
fast
good optimizer for real-valued optimisation
relatively much theory
Special:
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self-adaptation of (mutation) parameters standard
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
ES technical summary tableau
Representation
Real-valued vectors
Recombination
Discrete or intermediary
Mutation
Gaussian perturbation
Parent selection
Uniform random
Survivor selection
(,) or (+)
Specialty
Self-adaptation of mutation
step sizes
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Introductory example
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Task: minimimise f : R  R
Algorithm: “two-membered ES” using
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Vectors from Rn directly as chromosomes
Population size 1
Only mutation creating one child
Greedy selection
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Introductory example: pseudocde
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Set t = 0
Create initial point xt =  x1t,…,xnt 
REPEAT UNTIL (TERMIN.COND satisfied) DO
Draw zi from a normal distr. for all i = 1,…,n
yit = xit + zi
IF f(xt) < f(yt) THEN xt+1 = xt
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ELSE xt+1 = yt
FI
Set t = t+1
OD
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Introductory example: mutation
mechanism
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z values drawn from normal distribution N(,)
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 is varied on the fly by the “1/5 success rule”:
This rule resets  after every k iterations by
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mean  is set to 0
variation  is called mutation step size
 =  / c if ps > 1/5
 =  • c if ps < 1/5
=
if ps = 1/5
where ps is the % of successful mutations, 0.8  c  1
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Illustration of normal distribution
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Another historical example:
the jet nozzle experiment
Task: to optimize the shape of a jet nozzle
Approach: random mutations to shape + selection
Initial shape
Final shape
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Another historical example:
the jet nozzle experiment cont’d
Jet nozzle: the movie
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
The famous jet nozzle experiment (movie)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Genetic operators: mutations (2)
The one
dimensional case
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Representation
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Chromosomes consist of three parts:
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Object variables: x1,…,xn
Strategy parameters:
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Mutation step sizes: 1,…,n
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Rotation angles: 1,…, n
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Not every component is always present
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Full size:  x1,…,xn, 1,…,n ,1,…, k 
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where k = n(n-1)/2 (no. of i,j pairs)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutation
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Main mechanism: changing value by adding
random noise drawn from normal distribution
x’i = xi + N(0,)
Key idea:
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 is part of the chromosome  x1,…,xn,  
 is also mutated into ’ (see later how)
Thus: mutation step size  is coevolving with
the solution x
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutate  first
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Net mutation effect:  x,     x’, ’ 
Order is important:
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Rationale: new  x’ ,’  is evaluated twice
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first   ’ (see later how)
then x  x’ = x + N(0,’)
Primary: x’ is good if f(x’) is good
Secondary: ’ is good if the x’ it created is good
Reversing mutation order this would not work
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutation case 1:
Uncorrelated mutation with one 
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Chromosomes:  x1,…,xn,  
’ =  • exp( • N(0,1))
x’i = xi + ’ • N(0,1)
Typically the “learning rate”   1/ n½
And we have a boundary rule ’ < 0  ’ = 0
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutants with equal likelihood
Circle: mutants having the same chance to be created
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutation case 2:
Uncorrelated mutation with n ’s
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Chromosomes:  x1,…,xn, 1,…, n 
’i = i • exp(’ • N(0,1) +  • Ni (0,1))
x’i = xi + ’i • Ni (0,1)
Two learning rate parmeters:
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’ overall learning rate
 coordinate wise learning rate
  1/(2 n)½ and   1/(2 n½) ½
And i’ < 0  i’ = 0
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutants with equal likelihood
Ellipse: mutants having the same chance to be created
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutation case 3:
Correlated mutations
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Chromosomes:  x1,…,xn, 1,…, n ,1,…, k 
where k = n • (n-1)/2
and the covariance matrix C is defined as:
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cii = i2
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cij = 0 if i and j are not correlated
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cij = ½ • ( i2 - j2 ) • tan(2 ij) if i and j are correlated
Note the numbering / indices of the ‘s
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Correlated mutations cont’d
The mutation mechanism is then:
 ’i = i • exp(’ • N(0,1) +  • Ni (0,1))
 ’j = j +  • N (0,1)
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x ’ = x + N(0,C’)
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x stands for the vector  x1,…,xn 
C’ is the covariance matrix C after mutation of the  values
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  1/(2 n)½ and   1/(2 n½) ½ and   5°
i’ < 0  i’ = 0 and
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| ’j | >   ’j = ’j - 2  sign(’j)
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A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Mutants with equal likelihood
Ellipse: mutants having the same chance to be created
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Recombination
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Creates one child
Acts per variable / position by either
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Averaging parental values, or
Selecting one of the parental values
From two or more parents by either:
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Using two selected parents to make a child
Selecting two parents for each position anew
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Names of recombinations
Two parents
Two fixed parents
selected for each i
zi = (xi + yi)/2
Local
intermediary
Global
intermediary
zi is xi or yi
chosen randomly
Local
discrete
Global
discrete
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Parent selection
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Parents are selected by uniform random
distribution whenever an operator needs
one/some
Thus: ES parent selection is unbiased - every
individual has the same probability to be
selected
Note that in ES “parent” means a population
member (in GA’s: a population member
selected to undergo variation)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Survivor selection
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Applied after creating  children from the 
parents by mutation and recombination
Deterministically chops off the “bad stuff”
Basis of selection is either:
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The set of children only: (,)-selection
The set of parents and children: (+)-selection
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Survivor selection cont’d
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(+)-selection is an elitist strategy
(,)-selection can “forget”
Often (,)-selection is preferred for:
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Better in leaving local optima
Better in following moving optima
Using the + strategy bad  values can survive in x, too long
if their host x is very fit
Selective pressure in ES is very high (  7 •  is the
common setting)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Self-adaptation illustrated
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Given a dynamically changing fitness
landscape (optimum location shifted every 200
generations)
Self-adaptive ES is able to
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follow the optimum and
adjust the mutation step size after every shift !
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Self-adaptation illustrated cont’d
Changes in the fitness values (left) and the mutation step sizes (right)
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Prerequisites for self-adaptation
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 > 1 to carry different strategies
 >  to generate offspring surplus
Not “too” strong selection, e.g.,   7 • 
(,)-selection to get rid of misadapted ‘s
Mixing strategy parameters by (intermediary)
recombination on them
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Example application:
the cherry brandy experiment
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Task to create a colour mix yielding a target colour (that
of a well known cherry brandy)
Ingredients: water + red, yellow, blue dye
Representation:  w, r, y ,b  no self-adaptation!
Values scaled to give a predefined total volume (30 ml)
Mutation: lo / med / hi  values used with equal chance
Selection: (1,8) strategy
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Example application:
cherry brandy experiment cont’d
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Fitness: students effectively making the mix
and comparing it with target colour
Termination criterion: student satisfied with
mixed colour
Solution is found mostly within 20 generations
Accuracy is very good
A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing
Evolution Strategies
Example application:
the Ackley function (Bäck et al ’93)
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The Ackley function (here used with n =30):
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1 n 2
f ( x)  20  exp   0.2
  xi
n i 1
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1 n
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  exp   cos( 2xi )   20  e
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 n i 1
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Evolution strategy:
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Representation:
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-30 < xi < 30 (coincidence of 30’s!)
30 step sizes
(30,200) selection
Termination : after 200000 fitness evaluations
Results: average best solution is 7.48 • 10 –8 (very good)